I found a work-around: Tableau Online Sync Client!
What this little piece of gold basically does: It runs the refresh anywhere (e.g. my Laptop) and pushes it to Tableau Online. Exactly what I was looking for to avoid the Error mentioned above...
Some API with the data
-> WDC running on my Laptop to get the data into Tableau Format (Tableau Desktop 64 bit needs to be installed)
-> Tableau Online Client Sync scheduling automatic refreshes and sending the data to Tableau Online
One note, though: This works with a self developed WDC where you can bypass the authentication. I tried Client Sync with my WDC and the Tableau-stored credentials, but then Client Sync crashes. If I hard-code the credentials into the WDC so Client Sync doesn't have to authenticate at all it works fine
Thanks everyone who gave a thought into this
Oh, btw: I'm running both the WDC and Sync Client on my laptop because it is switched on most of the time anyway... Obviously this could run on a Windows 64 bit Server anywhere, too